Please give examples of the similarities and differences between the negative form of a proposition

The negation of a proposition is the same as the negation of a proposition.

A proposition is completely antithetical to its negative form. There is only one and only one between the two.

In mathematics, the method of counterproof is often used, and in order to prove a proposition, it is only necessary to prove that its negative form is not true.

How to get the negative form of a proposition? If you studied mathematical logic, it would be easy to understand, but now you can only understand it like this:

Original proposition: The square of all natural numbers is positive.

The standard form of the original proposition: any x, (if x is a natural number, then x is a positive number).

"Arbitrary" is a qualifier, "x is a natural number" is a condition, and "x is a positive number" is a conclusion. To deny a proposition, one needs to deny both its qualifier and its conclusion. The qualifiers "arbitrary" and "existent" negate each other.

Negative form: not (any x, (if x is a natural number, then x is a positive number)) x exists, (if x is a natural number, then x is not a positive number).

To put it another way: at least one natural number is not squared positively.

Whereas, the negative proposition of a proposition is used less. Whether a proposition is true or not has nothing to do with whether it is true or not.

It's easy to get a negative proposition for a question, just deny the qualifiers, conditions, and conclusions.

Original proposition: The square of all natural numbers is positive.

The standard form of the original proposition: any x, (if x is a natural number, then x is a positive number).

Negative proposition: x exists, (if x is not a natural number, then x is not a positive number).

To put it another way: there is an unnatural number whose square is not positive.

To put it simply, the negation of a proposition only negates the conclusion of that proposition, while the negation of a proposition negates the conditions and conclusions of the original proposition. For example: "If a>0."

Then the negation of the proposition a+b>0" is "if a>0."then a+b<=0" the negative proposition is "if a<=0, then a+b<=0".

The negation of the proposition is: p, not q

No proposition: non-p, non-q

Propositional negation is the same as negation, e.g.: If I'm rich, then Luffy is a pig. The negative proposition is, if I don't have money, then Luffy is not a pig. Negation, the form of negation, is that if I had money, then Luffy was not a pig.

The negative form of the proposition is as follows:

The proposition: p:vx>0, ()1 2 is in the negative form .

Answer analysis: porcelain home 3x, > 0(g)*21。

Analysis] according to the full name proposition"vx∈m,p(x)"The negation of the proposition is "3x." ∈m, p(x)"and the results will be obtained.

Explanation] Because the negation of the universal proposition is a special proposition, when negating the full proposition, one is to rewrite the full pronoun quantifier as an existential quantifier, and the other is to negate the conclusion, so the proposition p: vx > 0, (< 1 negation of p is x. > 0,()21。

So the answer is: 3x, >0,(g)*°l.

This question mainly examines the negation of the full proposition, which is a simple question: the negation of the full proposition and the special proposition is different from the negation of the proposition.

words, the full quantifier is rewritten as the existential quantifier, and the existential quantifier is rewritten as the full quantifier; The second is to negate the conclusion, and the negation of the general proposition only needs to directly negate the conclusion.

In modern philosophy, mathematics, logic, and linguistics, a proposition (judgment) refers to the semantics of a judgment sentence (the concept of actual expression), which is a phenomenon that can be defined and observed.

The proposition does not refer to the judgment sentence itself, but to the semantics expressed. When different judgments have the same semantics, they express the same proposition. In mathematics, it is generally called a proposition to judge a certain thing.

The negation of propositions is mainly aimed at simple propositions (ordinary propositions) and propositions containing quantifiers, and the rules of negation propositions of the original proposition are: negating the conclusion, and "replaceing" the quantifier, that is, replacing the full quantifier (existential quantifier) in the original proposition with the existential quantifier (full quantifier).This kind of proposition generally has only the negation of the proposition, but not the negation of the proposition.

The negative proposition of the original proposition: The original proposition at this point refers specifically to a proposition of the form "if p, then (then) q", and its negative proposition is "if it is not p, then (then) not q".Such a negation of the original proposition is likewise a negation of the conclusion only, i.e., the negation of the original proposition is:

If p, then (then) not q".

Note: The negation of a proposition and the negation of a proposition are for different types of original propositions, which are two different concepts.

A proposition is a linguistic representation of a situation or concept in the real world, which can be true or false. A negative proposition, on the other hand, is a statement of the opposite of a proposition, i.e., a reversal of its truth or falsehood. In contrast to this is the negation of propositions, which is not a simple reversal of the concept of "negation of propositions", but the negation of a certain qualifier in a proposition, which can be either positive or negative.

For example, if there is a proposition "A is a meeting that is popular", then to deny that proposition is "the conference is not popular", which is a negation of the whole proposition. But if the proposition is "the meeting held by A is very popular because the venue is spacious and bright", then the denial of it is "the meeting held by A is not popular because the venue is small and dim", which is the negation of the condition "the venue is spacious and bright".

When we deny a proposition, it becomes an opposite proposition. For example, "The meeting organized by A is popular" and "The meeting organized by A is not popular" are two opposite propositions. Since there are two opposite states, "very popular" and "unpopular", these two propositions are directly negative to each other.

In mathematical logic, a negative proposition is often expressed directly as "non-p", indicating the negation of the proposition p. The negation of the proposition p is expressed as "the negation of p" or "not p". In logical reasoning, the relationship between the negation of propositions and the negation of propositions can help us better understand and analyze the truth or falsity of statements and help us better understand the process of reasoning and argumentation.

In summary, although the negative proposition judgment and the negation of the proposition are similar in concept, they are actually different. A negative proposition is a statement that turns the entire proposition upside down and expresses the opposite meaning. The negation of a proposition is the negation of a certain qualifier in a proposition, which is a specific logical operation.

Through the understanding of these two concepts, we can better conduct propositional analysis and logical reasoning. <>

The negative proposition of the original proposition is to negate all the conditional conclusions, and the negative proposition is more complicated, and it is generally sufficient to grasp the negation of the special proposition and the full proposition.

If p then the negation of the q form can be ignored, and the negation of a or b is. The negation of non-A and non-Ba and B is. Non-A or Non-B

In the case of negation, "and" and "or" need to be interchanged.

The difference between the negation of a proposition and the negation of a proposition.

The negation or negation of a proposition is a completely different concept. The reasons for this are: first, there is a negation of any proposition, whether it is a true proposition or a false proposition; The negative proposition is only raised for the proposition "if p then q".

Second, the negation of the proposition is the contradictory proposition of the original proposition, and the truth and falsity of the two must be one true and one false, one false and one true; The negative proposition and the original proposition may be the same true and false, or they may be the same true and one false. As can be seen from the truth table below: pq

pq "pqpq" three, the original proposition "if p then q".

form, its negation proposition has been said earlier; And its negative proposition is "if it is not p, then it is not q", (denoted as "if p, then q"), that is, it negates both the condition and the conclusion.

Example 6: Write the negative or negative proposition of the following proposition. and judge its authenticity.

If x y, then 5x 5y.

If x2+x 2 is quietly enabled, then x2-x 2.

The four sides of the square are equal.

It is known that a,b are real numbers, and if x2+ax+b 0 has a non-empty real solution set, then a2-4b 0.

Solution: (1) The negation:

x,y (x y and 5x 5y).

False propositions. No proposition: v

x,y（x≤y

5x≤5y）。

True proposition. The original proposition is: v

x,y（x＞y

5x＞5y）。True proposition.

2) The negation:

x(x2+x2, and x2-x2). True proposition.

Negative proposition: vx(x2+x 2, x2-x 2). False propositions.

The original proposition is: v

x（x2+x﹤2,x2-x﹤2）。False propositions.

3) Negation: There is a quadrilateral, and although it is a square, at least two of the four sides are not equal. False propositions.

No proposition: If a quadrilateral is not a square, then its four sides are not equal. False propositions.

The original proposition is the true proposition.

See example 5(5)).

4) Negation: There are two real numbers a, b, although satisfying x2+ax+b 0 has a non-empty set of real solutions, but makes a2-4b 0. False propositions.

Negative proposition: Knowing that a, b are real numbers, if x2+ax+b 0 does not have a non-empty real solution set, then a2-4b 0. True proposition.

The original proposition is: for any real number a, b, if x2+ax+b 0 has a set of non-empty real solutions, then a2-4b 0 true proposition).

In teaching, it is necessary to clarify the formal structure and nature relationship of various types of propositions. Only then can we truly and accurately express the negation of propositions, and if we can avoid logical errors, we can better load logical knowledge on other knowledge, so as to cultivate and develop students' logical thinking ability.